This paper applies a recently developed technique for deriving long-term
trends in ozone from sparsely sampled data sets to multiple occultation
instruments simultaneously without the need for homogenization. The technique
can compensate for the nonuniform temporal, spatial, and diurnal sampling of
the different instruments and can also be used to account for biases and
drifts between instruments. These problems have been noted in recent
international assessments as being a primary source of uncertainty that
clouds the significance of derived trends. Results show potential
“recovery” trends
of ∼ 2–3 % decade−1 in the upper stratosphere at
midlatitudes, which are similar to other studies, and also how sampling
biases present in these data sets can create differences in derived recovery
trends of up to ∼ 1 % decade−1 if not properly accounted for.
Limitations inherent to all techniques (e.g., relative instrument drifts) and
their impacts (e.g., trend differences up to ∼ 2 % decade−1)
are also described and a potential path forward towards resolution is
presented.

Ever since the Montreal Protocol came into effect, the global
scientific community has been monitoring the state of stratospheric ozone in
an effort to determine at first if the loss rate was decreasing and later if
ozone had begun to recover. Consequently, there has been an ongoing body of
work to use single (at first) or multiple (later) sources of data, spanning
the satellite record starting around 1979, for various multiple linear
regression (MLR) analyses to determine the long-term trends in stratospheric
ozone. A simple literature search would reveal the various techniques and
studies ranging from the earlier works
(e.g., Wang et al., 1996; Bodeker et al., 1998; Newchurch et al., 2003) revealing the loss
slowdown to a recent surge in efforts to determine potential ozone recovery
(e.g., Randel and Wu, 2007; Remsberg and Lingenfelser, 2010; Bodeker et al., 2013; Kyrölä et al., 2013; Bourassa et al., 2014; Gebhardt et al., 2014; Tummon et al., 2015; Harris et al., 2015; Steinbrecht et al., 2017).
These works have culminated in the most recent Scientific Assessment of Ozone
Depletion (WMO, 2014) that showed statistically significant “recovery”
trends of ∼ 2 % decade−1 in the upper stratosphere at
midlatitudes but identified three factors with a potential major impact that were
not readily accounted for in those analyses: diurnal variability of ozone,
biases between data sets, and long-term drifts between data sets. There is an
additional complication that is intricately tied to these three factors in
this kind of analysis, namely the nonuniform temporal, spatial, and diurnal
sampling of the different instruments used for these analyses. This
nonuniform sampling can have a detrimental impact not only on the regression
techniques used to derive long-term trends in ozone but also on other
analyses performed to determine diurnal variability or the magnitude of
potential biases and drifts between data sets. Herein, we discuss a recently
developed technique that not only accounts for the potential sampling issues,
but also the perceived diurnal variability, as well as any potential bias
and/or drift between instruments in a single analysis.

There have been several remote sensing instruments over the past several
decades that have observed stratospheric ozone using the method of solar
occultation, including but not limited to the Atmospheric Chemistry
Experiment Fourier transform spectrometer (ACE-FTS); the Halogen Occultation
Experiment (HALOE); the Polar Ozone and Aerosol Measurement (POAM) II
and III; and the Stratospheric Aerosol and Gas Experiment (SAGE) I, II,
and III. For the purpose of this study, however, SAGE I was ignored because
it does not have any overlap with the other missions.

2.1 ACE-FTS

The Atmospheric Chemistry Experiment Fourier transform spectrometer (ACE-FTS)
was launched onboard the SCISAT-1 spacecraft in August 2003
(Bernath, 2017). The spacecraft occupies a 74∘ inclined orbit at an
altitude of ∼ 650 km that allows for observations
from 85∘ S to 85∘ N. The primary ACE instrument is a high
spectral resolution (0.02 cm−1) Fourier transform
spectrometer operating in the spectral range
of ∼ 2.2–13.3 µm
(750–4400 cm−1) that measures many trace gas species
and isotopologues (Bernath et al., 2005). Ozone is retrieved using the
spectral features near 10 µm(Boone et al., 2005). The
version of the ACE-FTS data product used here is version 3.5
(Boone et al., 2013), which produces vertical profiles of volume mixing
ratio (VMR) interpolated to a 1 km grid with a vertical resolution
of 3–4 km. The ACE-FTS instrument is still operating.

2.2 HALOE

The Halogen Occultation Experiment (HALOE) was launched onboard the Upper
Atmosphere Research Satellite (UARS) in September 1991. The spacecraft
occupied a 57∘ inclined orbit at an altitude
of ∼ 585 km that allowed for observations
from 80∘ S to 80∘ N. The HALOE instrument used a combination of
broadband radiometry and gas filter correlation techniques to observe several
trace gas species in the spectral range
of ∼ 2.4–10.4 µm
(∼ 950–4150 cm−1) and measured
ozone using the spectral band near 9.6 µm(Russell et al., 1993). The version of the HALOE data product used here is
version 19.0 (Thompson and Gordley, 2009), which produces vertical profiles
of VMR interpolated to a 0.3 km grid with a vertical resolution of
2–3 km(Bhatt et al., 1999). The UARS mission was
decommissioned in December 2005.

2.3 POAM II

The Polar Ozone and Aerosol Measurement II (POAM II) was launched onboard the
SPOT-3 spacecraft in September 1993. The spacecraft occupied a
sun-synchronous orbit, crossing the descending node at 10:30 LT, that allowed
for observations in two latitude bands at 88 to 62∘ S
and 65 to 71∘ N. The POAM II instrument used broadband
radiometry to observe aerosol and trace gases in the spectral range
of ∼ 350–1070 nm and measured ozone using the
spectral band near 600 nm(Glaccum et al., 1996). The version of
the POAM II data product used here is version 6.0, which produces vertical
profiles of number density interpolated to a 1 km grid with a
vertical resolution of 1 km(Lumpe et al., 1997). The SPOT-3
spacecraft ceased functioning in November 1997.

2.4 POAM III

The Polar Ozone and Aerosol Measurement III (POAM III) was launched onboard
the SPOT-4 spacecraft in March 1998. The spacecraft occupied a
sun-synchronous orbit, crossing the descending node at 10:30 LT, that allowed
for observations in two latitude bands at 88 to 62∘ S
and 65 to 71∘ N. The POAM III instrument used broadband
radiometry to observe aerosol and trace gases in the spectral range
of ∼ 345–1030 nm and measured ozone using the
spectral band near 600 nm(Lucke et al., 1999). The version of
the POAM III data product used here is version 4.0
(Lumpe et al., 2002; Naval Research Laboratory, 2006), which produces vertical
profiles of number density interpolated to a 1 km grid with a
vertical resolution of 1 km. The POAM III instrument ceased
functioning in December 2005.

2.5 SAGE II

The Stratospheric Aerosol and Gas Experiment II (SAGE II) was launched
onboard the Earth Radiation Budget Satellite (ERBS) in October 1984. The
spacecraft occupied a 57∘ inclined orbit at an altitude
of ∼ 610 km that allowed for observations
from 80∘ S to 80∘ N. The SAGE II instrument was a broadband
spectrometer that operated in the spectral range
of ∼ 375–1030 nm for aerosol and trace gas
observations and measured ozone using the spectral band near 600 nm(Mauldin III et al., 1985). The version of the SAGE II data product used here
is version 7.00 (Damadeo et al., 2013), which produces vertical profiles of
number density interpolated to a 0.5 km grid with a vertical
resolution of 1 km. The ERBS mission was decommissioned in
October 2005.

2.6 SAGE III

The Stratospheric Aerosol and Gas Experiment III (SAGE III) was launched
onboard the Russian Meteor-3M (M3M) spacecraft in December 2001. The
spacecraft occupied a sun-synchronous orbit, crossing the ascending node
at 09:00 LT, that allowed for observations in two latitude bands at 60
to 30∘ S and 45 to 80∘ N. The SAGE III instrument was
a grating spectrometer that operated in the spectral range
of ∼ 295–1025 nm for aerosol and trace gas
observations and measured ozone using the spectral features
near 600 nm(Mauldin et al., 1998). The version of the SAGE III
data product used here is version 4.00
(Cunnold and McCormick, 2002; Wofsy et al., 2002), which produces
vertical profiles of number density interpolated to a 0.5 km grid
with a vertical resolution of 1 km. The M3M spacecraft ceased
functioning in January 2006.

2.7 Filtering

When making use of any data set, it is important to apply the proper
filtering to ensure that bad data (e.g., fill values or data contaminated by
clouds) are excluded. Since this analysis is constrained to the stratosphere,
all data below the tropopause are ignored. If a data set provides a
tropopause height, that value is used for filtering purposes, otherwise the
World Meteorological Organization (WMO) definition is used (WMO, 1992).
Beyond this, the data screening procedures recommended for each data set are
performed. ACE-FTS data are screened as outlined in Sheese et al. (2015).
HALOE data are screened for potential problematic “constant lockdown angle”
and “trip angle” events as detailed by the data producers
(http://haloe.gats-inc.com/user_docs/index.php). POAM II data could be
screened for interference from polar stratospheric clouds (PSCs) by looking
for outliers in the 1 µm data, though this is not performed.
POAM III data are screened for potential sunspot interference and heavy
aerosol interference through the use of the quality flags. SAGE II data are
screened for this analysis in the same way as was done in
Damadeo et al. (2014). Since SAGE III data were screened prior to release, no
additional screening is performed.

In principle, this work is a continuation of the work first performed in
Damadeo et al. (2014) and so the same techniques and methodologies are used.
Each data set is filtered according to the stated filtering techniques and
converted to the unit system of interest (i.e., number density or mixing
ratio versus altitude or pressure) using the pressures, temperatures, and
altitudes provided with the respective data sets. While we did apply the
analysis to both combinations of unit systems, for the sake of brevity all
results shown here are for regressions to data in number density on altitude
(some mixing ratio on pressure results are shown in the Supplement).
Additionally, the data for each instrument were interpolated to
0.5 km increments. These data are then consolidated into daily zonal
means for each instrument separated by both satellite and local event types.
A generalized least-squares regression technique that accounts for
autocorrelation, heteroscedasticity, and data gaps is then performed on all
data sets simultaneously, with the autocorrelation and heteroscedasticity
corrections being applied separately for each instrument. In principle, this
technique is applicable to data sets with higher sampling (e.g., the
Microwave Limb Sounder, MLS, on the Aura satellite) but is demonstrated here
on occultation data sets only to illustrate the impact of their sparse
sampling patterns on derived trends.

Figure 1Location of all SAGE III occultation events for both spacecraft
(a) and local (b) event types. In each case, sunrises are shown in
blue while sunsets are shown in red. While there is a clear hemispheric
distinction between spacecraft event types, nearly all local event types are
sunsets with the exception of spacecraft sunset events in polar winter. Other
occultation instruments in sun-synchronous orbits such as POAM II and
POAM III exhibit similar behavior.

The same simultaneous temporal and spatial (STS) MLR model as was used in
Damadeo et al. (2014) is applied using the same proxy terms, albeit with nine
spatial terms instead of seven and some additional changes to account for the
incorporation of multiple instruments (see Appendix A). Terms
accounting for the quasi-biennial oscillation (QBO), El Niño–Southern
Oscillation (ENSO), solar variability, and long-term trends (two orthogonal
equivalent effective stratospheric chlorine, EESC, functions) are applied to
all data sets simultaneously. Terms accounting for volcanic eruptions
(primarily the Mount Pinatubo eruption in 1991) are applied to the SAGE II
and HALOE data sets only (and separately) to avoid potential overfitting of
minor eruptions in data sets that do not cover the Mt. Pinatubo eruption.
Diurnal variability (applied as a binary conditional term) is fit separately
for each data set. While the sun-synchronous instruments (i.e., SAGE III,
POAM II, and POAM III) sample both satellite event types, they do not
adequately sample both local event types (Fig. 1) and so all
local sunrises from these instruments are ignored for this analysis. The
seasonal cycle is applied to all data sets simultaneously as a single
seasonal cycle for all instruments. Lastly, a bias offset term and a linear
drift term are applied separately for each instrument using SAGE II as the
reference instrument.

Figure 2Spread of the correlated and uncorrelated residuals as a function of
latitude and altitude for each instrument from the regression. White regions
show areas where insufficient data exist.

Since the STS regression model uses a two-dimensional regression, it is best
utilized on data that adequately cover the full range of temporal and
spatial sampling to constrain the temporal and spatial variability present in
the data. Occultation instruments in mid-inclination orbits tend to deliver
near-global coverage at somewhat reduced seasonal sampling while occultation
instruments in sun-synchronous orbits tend to deliver highly localized
spatial coverage at nearly full seasonal sampling. The primary focus of this
work is the impact of sampling biases on long-term trends in ozone, which is
typically analyzed in the stratosphere between about 60∘ S
and 60∘ N. Since this work focuses on that latitude range and the
sun-synchronous instruments exhibit little to no coverage within that region
and thus very little influence on resulting trends there, the results
presented herein derive from an STS regression using only the SAGE II, HALOE,
and ACE-FTS data sets. We also applied an STS regression using all six data
sets and found that the long-term trends were not significantly affected (see
the Supplement) but did notice that the lack of spatial coverage in the
POAM II, POAM III, and SAGE III data sets detrimentally impacted the results
in the seasonal cycle and diurnal variability derived from a two-dimensional
regression. In the interest of brevity and to maintain the legibility of
certain figures in this paper, individualized results from the six-instrument
regression are not shown here.

4.1 Residuals

Similarly to Damadeo et al. (2014), we investigate the residuals of the
regression. The residuals from the regression can be used to ascertain the
quality of the model and the data set itself, independent of any offset in
the mean value. While the mean of the residuals is zero (as it should be), a
clear pattern in the spread of the residuals emerges as a function of
latitude at each altitude. The total residuals of the regression (i.e., the
residuals from the ordinary least-squares regression) are a combination of the
correlated residuals (i.e., those removed during the autocorrelation
correction) and the uncorrelated residuals (i.e., the residuals from the
generalized least-squares regression). The correlated residuals represent
geophysical variability that is well sampled but not well modeled by the
regression as well as any systematic instrumental variability (e.g., biased
meteorological or ephemeris input data). The uncorrelated residuals represent
both measurement noise and geophysical variability that is not well sampled
(e.g., geophysical variability present within each daily mean).

Figure 2 shows the spread of the correlated and
uncorrelated residuals for each instrument. All of the instruments exhibit
increased residuals in the lower stratosphere, owing both to the increased
uncertainty of measurements in that region as well as increased variability
that is not adequately captured by the proxies used for this regression.
Similarly, residuals are higher at higher latitudes where measurements can
routinely dip into and out of the vortex both over multiple days and within a
single day itself. SAGE II has greatly increased uncorrelated residuals at
the highest altitudes compared to HALOE and ACE-FTS. While the influence of
measurement noise and daily zonal variability in the uncorrelated residuals
cannot be separated, the fact that SAGE II and HALOE (and to a lesser extent
ACE-FTS) exhibit similar sampling patterns means that the increased
uncorrelated residuals in the upper stratosphere and lower mesosphere in
SAGE II compared to HALOE must be a result of increased measurement noise in
SAGE II. Similarly, SAGE II and ACE-FTS display slightly lower uncorrelated
residuals in the lower stratosphere while HALOE and ACE-FTS display lower
uncorrelated residuals in the upper stratosphere. All three instruments show
comparable uncorrelated residuals in the middle stratosphere.

The correlated residuals show an increased spread in the stratosphere at high
latitudes, which is expected as variability within the polar vortex is not
modeled in this regression. Similarly, increases can be seen in the tropical
middle stratosphere near a local peak in QBO amplitude. This is a result of a
two-dimensional fit using a proxy derived only at the Equator. While
modulating the QBO with the seasonal cycle better represents the QBO at
higher latitudes, the inability to accurately model the QBO at higher
latitudes detracts from the ability to accurately model the QBO at lower
latitudes (Damadeo et al., 2014). Another interesting feature is an apparent
vertical banding structure in the correlated residuals present in each data
set. The locations of this banding correlate to the turnover latitudes in
each instrument's orbit (i.e., the latitudes at which measurements go from
progressively closer to the poles to progressively further away). The
autocorrelation correction accounts for the degree of correlation of data
from day to day. However, the locations of daily means change in latitude
from day to day, with rates of motion greater at the Equator and smaller near
the poles, and so the degree of correlation is dependent upon both the
temporal variability and the meridional variability, with the meridional
variability being the primary driver. At the orbit turnover point, the
meridional variability between each successive daily mean essentially
disappears. While not explored in this study, it is possible that this
additional source of correlated noise stems from the nature of how wave one
action is sampled from day to day over the course of about 1 week until the
instrument moves away from the turnover latitude. Because the measurements
systematically shift in longitude over the day while the wave itself also
rotates, the zonal variability is not evenly sampled and so these day to day
differences will be highly correlated as the wave one action rotates and
changes, thus revealing a potential additional source of sampling bias albeit
more localized and on a shorter timescale.

4.2 Diurnal variability

Occultation instruments sample one sunrise (SR) and one sunset (SS) per orbit as seen
by the spacecraft, which typically correlates to one sunrise and one
sunset as seen by an observer on the ground at the measurement location.
This means that occultation measurements of ozone sample its diurnal
variability present in the mesosphere and upper stratosphere. Diurnal
variability of ozone in the mesosphere has been investigated before and is
well understood to be a result of rapid photochemistry across the terminator
(Chapman, 1930; Herman, 1979; Pallister and Tuck, 1983). While the full
attribution of sources is still not completely understood, diurnal
variability in the stratosphere is well represented in various data sets.
Analysis of the diurnal variability from occultation instruments is typically
performed by looking for periods where the instrument's diurnal sampling
“crosses itself” (i.e., local sunrise and sunset measurements occur at
roughly the same latitude at roughly the same time). Sakazaki et al. (2015)
used this method to analyze the diurnal variability present in SAGE II,
HALOE, and ACE-FTS and found that not all data sets agree and the differences
between SR and SS values differ typically by up to ∼ 5 %.
The STS regression can extract the mean diurnal variability present in each
data set and the results shown in Fig. 3 compare quite well
with those in Fig. 5 of Sakazaki et al. (2015).

Figure 3Results from the regression depicting the mean diurnal variability
present in each data set plotted as the percent difference between sunrise
and sunset events. These results compare well with those of
Sakazaki et al. (2015).

4.3 Impacts of aerosol

Volcanic eruptions periodically inject sulfur dioxide into the stratosphere
where it goes on to form sulfate aerosols that can impact ozone either via
chemical effects (Rodriguez et al., 1991; Solomon, 1999) or through changes in
dynamics via changes in radiative forcing (McCormick et al., 1995; Robock, 2000).
In either case, it is possible for volcanic aerosols to have a significant
impact on stratospheric ozone levels such that their presence can complicate
these regression analyses. Since ozone trend analyses utilize data from the
past ∼ 30 years, usually only the Mt. Pinatubo
eruption in mid-1991 is considered for special treatment. If the analysis
goes back further, sometimes the El Chichón eruption in early 1982 is also
considered. The punctuated nature of the eruptions and not completely
characterized impacts on data quality often leads to many works simply
excluding data from 1 to several years after these eruptions
(e.g., Wang et al., 1996; Randel and Wu, 2007; Harris et al., 2015), while some works attempt
to include a term in the regression to model the impact
(e.g., Bodeker et al., 2001; Stolarski et al., 2006; Bodeker et al., 2013; Tummon et al., 2015),
although the nature of these terms tends to be different between different
analyses.

For this work we include an aerosol proxy that was derived in
Damadeo et al. (2014). The proxy is a volcanic one, meaning that eruptions
occur and the proxy rises, peaks, and subsequently decays back to zero. The
proxy only covers the SAGE II mission time period and thus is zero throughout
most of the ACE-FTS mission period. However, given that it takes a relatively
large eruption (e.g., Mt. Pinatubo) to register any noticeable changes in
stratospheric ozone in these regression analyses and the fact that only minor
eruptions have occurred since (Vernier et al., 2011), this is assumed to be
sufficient. Given that occultation instruments can (depending upon their
spectral channels) have reduced measurement sensitivity in the presence of
heavy aerosol loading
(Wang et al., 2002; Bhatt et al., 1999), the volcanic proxy
is applied separately for SAGE II and HALOE. The regression was applied under
two conditions with regard to aerosol: one in which no filtering of events
for the influence of aerosols was performed and another in which SAGE II was
filtered under the recommendations in Wang et al. (2002) and
HALOE was filtered under the recommendations in
Bhatt et al. (1999).

Figure 4Peak of the volcanic term near the eruption of Mt. Pinatubo as a
percentage of the local mean for both SAGE II and HALOE under different
regressions. Results for SAGE II are shown both with and without the
Wang et al. (2002) filtering criteria. Results for HALOE are
shown without any aerosol filtering, though results with filtering are
similar.

Figure 4 shows the peak of the volcanic regression term
surrounding the Mt. Pinatubo eruption for both the aerosol filtered and
unfiltered cases. In the unfiltered case, both SAGE II and HALOE show similar
responses of ozone to the eruption in the tropics
between ∼ 24 and 35 km. Both instruments show a
large region of negative correlation between ozone and aerosol in the lower
stratosphere surrounding the aerosol layer itself and another large region of
positive correlation in the middle stratosphere above the aerosol layer (the
anomalously large responses in the lowermost stratosphere are a result of
overfitting due to missing data). These results are in reasonably good
agreement with Aquila et al. (2013) and Bodeker et al. (2013), which show
results of the impact of the eruption on ozone levels from modeling and data
respectively, and in surprisingly good agreement between the two separate
instruments. The effect of the eruption on ozone derived from HALOE data is
typically more difficult to quantify since HALOE did not begin to take
measurements until shortly after the eruption, which has a tendency to
negatively impact studies of long-term variation using only the HALOE data
set (Remsberg, 2008).

When the regression was run with the stated aerosol filtering criteria
applied to the data, the results of the volcanic regression term from HALOE
remain unchanged (not shown). However, compared to the unfiltered case
(middle plot in Fig. 4) the SAGE II responses with aerosol
filtering applied (left plot in Fig. 4) remain unchanged
above 28 km but significantly reduced in amplitude in the tropics
below that. The aerosol filtering has no effect in the middle stratosphere in
the region of positive correlation because the aerosol loading levels were
not so high as to detrimentally affect the retrievals of these occultation
instruments there. Between the middle and the lowermost stratosphere the data
quality declined until measurements were no longer possible. The
Wang et al. (2002) filtering criteria were meant to exclude
anomalous ozone values based on aerosol extinction and aerosol extinction ratio values in the regions
where data quality declines. However, the apparent agreement of the
unfiltered results suggest that the Wang et al. (2002)
filtering criteria are overly conservative and need to be revisited. Either
that, or the SAGE II filtering criteria and results are reasonable and
perhaps the HALOE data require a better aerosol correction in the retrieval
algorithm than what is already applied (Hervig et al., 1995).

4.4 Solar cycle response

The impact of the ∼ 11-year solar cycle on stratospheric ozone has
been an ongoing topic of study
(e.g., Wang et al., 2002; Soukharev and Hood, 2006; Randel and Wu, 2007; Remsberg, 2008, 2014; Maycock et al., 2016; Dhomse et al., 2016).
As such, it is worthwhile to show the results of the solar response in this
work as well as to point out a few things about data usage and the
determination of the solar cycle response to ozone when using MLR-based
studies on SAGE II and HALOE data. The cited works show different solar
cycles when using SAGE II data as well as different solar cycles between
using SAGE II and HALOE data, with the latter exhibiting the greatest
difficulty in determining the solar cycle from only HALOE data
(Soukharev and Hood, 2006). Figure 5 shows the latitude- and
altitude-dependent amplitude of the solar cycle response derived from this
work, which is similar to other recent works based on the usage of the
SAGE II data set (e.g., Maycock et al., 2016; Dhomse et al., 2016) and naturally very
similar to those from Damadeo et al. (2014). One important distinction between
the previous work and this one is the impact of the use of one or two solar
terms. Previously, when applied to only SAGE II data, using two solar terms
shifted the solar cycle response by about 2 years in the presence of
the Mt. Pinatubo eruption in agreement with Remsberg (2014), though
this was believed to be the regression algorithm simply trying to attribute
some of the aerosol response to the solar cycle (Solomon et al., 1996). The
inclusion of HALOE and ACE-FTS in this study, however, seems to better
constrain the solar cycle such that using one or two solar cycle terms no
longer creates temporal shifting in the presence of the eruption (not shown)
and thus only a single solar cycle term is required for the regression. While
also not shown here, we attempted to apply the STS regression to only HALOE
data and found that no combination of proxies exhibited realistic-looking
solar cycle responses, most likely due to the data having insufficient
duration capable of constraining the solar cycle, aerosol, and trend terms
simultaneously. This could potentially explain the often different solar
cycle responses derived when using the instruments separately, while using
them simultaneously creates SAGE II-like responses as well as very similar
aerosol responses, though this requires further study. Lastly, it is worth
noting that the large amplitude tropical response
below ∼ 23 km is a result of the previously discussed
anomalous aerosol response in that area.

Figure 5Amplitude of oscillation of the solar cycle response as a percentage
of the local mean. Stippling denotes areas where the values are not
significant at the 2σ level. Contour lines are plotted at intervals of
0.5 %.

5.1 Seasonal sampling

Traditionally, data sets are reduced to monthly zonal mean (MZM) values for
regression analyses to determine long-term trends. Practically speaking,
these MZM values are utilized as though they are representative of the center
of the month and the center of the latitude bin. Though this assumption holds
mostly true for highly sampled data sets (e.g., nadir and limb sounders), it
generally fails when applied to occultation data sets. This fact is well
known and has been studied before. Toohey et al. (2013) and
Sofieva et al. (2014) both investigated nonuniform temporal sampling as an
added source of noise and uncertainty that could be characterized and
included in trend analyses. Using deseasonalized anomalies for trend analysis
can mitigate the impacts of sampling bias if the bias is constant with time.
However, owing to the observational geometry of occultation instruments and
orbital parameters (i.e., altitude, inclination, and precession rates) the
sampling patterns often tend to systematically drift over time as shown in
the top row of Fig. 6. Millán et al. (2016) investigated the
impacts of nonuniform sampling biases on resulting trends from different
instruments by using a “representative year” of sampling for each data set
and repeating it over 30 years to analyze the effect on trends. While
illustrative, this did not account for the actual sampling bias as it changed
from year to year.

Figure 6(a) The MZM temporal sampling bias shown as the difference
between the average time of sampling in a given month and latitude band and
the center of that month. Results shown here are for different months and
different data sets, though systematic biasing of results is common for most
months for each data set. (b) The MZM seasonal sampling bias shown as
the difference in ozone between the actual center of sampling for a given
month and latitude band and the center of that month and band as computed
using the seasonal cycle and the local mean from the STS regression. Results
are shown here for different altitudes illustrating the pervasiveness of the
problem.

The systematic drift in sampling combined with the presence of sampling
biases precludes the use of the MZM method to accurately determine the
seasonal cycle that is represented by an occultation data set. The STS
regression, however, is less sensitive to sampling biases and can thus be
used to quantitatively assess the sampling biases that would be present in
the MZM method. It is relatively straight forward to compute the temporal and
spatial offset between the average time and location of sampling within a
given month and latitude band and the center of that month and band that is
considered the representative location for the MZM method. The spatially
varying seasonal ozone cycle from the STS regression can then be used to
compute the difference in fitted ozone values between the actual center of
sampling and the representative center of sampling to compute a seasonal
sampling bias for each month, latitude band, and altitude bin. Some typical
results of these biases are shown in the bottom row of Fig. 6.
It is evident from a simple visual inspection of these results that drifting
sampling patterns create patterned monthly biases.

Figure 7(a) Yearly average of the MZM seasonal sampling biases
illustrated in the bottom row of Fig. 6. While the amplitude of
systematic biases decreases from the individual months, systematic biases are
still apparent. (b) Data extracted from the specified latitude band
in the top row are plotted in black in each case. The solar cycle (red) and
long-term trend (blue) from the STS regression for those altitudes and
latitudes are overplotted to illustrate the potential correlation between the
systematic sampling biases and long-duration variability.

While it is clear that these sampling biases will create problems attempting
to use the MZM method to assess the seasonal cycle or how it changes over
time, this investigation is more focused towards the effects on long-duration
variability. For each year, latitude band, and altitude bin, an average of the
monthly sampling biases can be computed to produce a yearly averaged bias
shown in the top row of Fig. 7. While the magnitudes of
yearly averaged biases are smaller than those of monthly biases, systematic
patterns are still evident. To illustrate the potential impact these sampling
biases can have when incorporated into regression analyses, we can look at an
individual sampling bias time series by extracting data from the top row of
Fig. 7 and plotting it along with the low frequency variability
from the STS regression. The bottom row of Fig. 7 shows this
data in black with the solar cycle (red) and long-term trend (blue)
overplotted to demonstrate how easily the drifting sampling patterns create
patterned biases that alias into interannual and long-term geophysical
variability. This will ultimately interfere with the ability of any analysis
to accurately determine the “true” long-term trends. While it may appear that
these results are being cherry-picked (and, since only so many figures can
be shown, they are), in actuality it is a “fruitful tree” and results shown
here are common (see the Supplement for more plots).

It should be noted that while the presence of seasonal sampling biases that
alias into longer-duration terms is pervasive (in altitude and latitude for
each data set), the actual degree of correlation with terms such as the solar
cycle or trend is somewhat more random as it is dependent upon the chance
combination of drifting sampling patterns, spatially varying seasonal
gradients, and frequency of interannual variability. Additionally, the
seasonal sampling biases will correlate with multiple terms simultaneously,
making a simple and concise quantitative evaluation of their impact on the
analysis results almost impossible. Ultimately, however, it is readily
apparent that the use of an MZM analysis method on data with obvious seasonal
sampling biases will produce biased results in derived long-term variability.

5.2 Diurnal sampling

Figure 8Monthly zonal sampling for SAGE II separated by local event type
(a, b). There was a problem with the battery that caused shortened
sunset events between mid-1993 and mid-1994 and an issue with the azimuthal
pointing system after late-2000 that caused a hemispheric asymmetry in
sampling. Panel (c) is the difference between the (a) and (b)
panels, revealing the rapid oscillation between SR- and SS-dominated months as
well as whole periods dominated by one event type.

With a few exceptions (e.g., Kyrölä et al., 2013; Remsberg, 2014; Damadeo et al., 2014),
most analyses of ozone trends make use of MZM values where SR and SS
measurements are treated equally. This has been done with the assumption that
the mean value will fall between the SR and SS means but that any sampling
biases are a source of random noise and do not affect the trend. As a result,
should the distribution of diurnal sampling not be evenly distributed, the
risk of a diurnal sampling bias becomes apparent.

In a similar way as the seasonal sampling, the nature of the orbit of the
spacecraft dictates how the instrument will sample local sunrises and sunsets
as a function of time of year and latitude over the mission lifetime. An
example of the diurnal sampling of the SAGE II instrument over its lifetime
is shown in Fig. 8. The most apparent features are the
increased rate of sampling at midlatitudes versus high and low latitudes and
the presence of instrument problems during the mission that caused asymmetric
diurnal sampling between mid-1993 and mid-1994 and after 2000. However, it is
by taking the difference between the sunrise and sunset sampling that the
true diurnal sampling differences become apparent. A close investigation of
the bottom panel of Fig. 8 for any given latitude
reveals a rapid oscillation of monthly biases between SR- and SS-dominant
months. In the presence of significant (i.e., a few percent) diurnal
variability such as in the upper stratosphere, this sampling bias will
interfere with the derivation of the seasonal cycle for an MZM analysis.

To get a better idea of the systematic long-term nature of the diurnal
sampling bias, we have looked at larger latitude bands (i.e., 35–45∘ N–S and 15∘ S–15∘ N) using the data in
Fig. 8. By smoothing the data over a year, we can dampen
some of the seasonal effects and more easily investigate the long-term
changes. Also, to intercompare different latitude bands, it is preferable to
look at the differences between SR and SS sampling as a percentage of the
total events rather than the absolute number of events. This actually raises
the question of whether to convert the differences as a number of events to a
percentage of the total number of events (for each month) and then smooth or
the other way around. It is interesting because this question draws a
corollary with the concept of computing unweighted or weighted monthly mean
values. If MZM values were computed by first calculating a mean value for
each month and then computing a mean July (for example) by simply taking the
mean of all Julys, that would be unweighted (i.e., all Julys are treated
equally in the overall mean regardless of how many events are in each July)
and would be analogous to our converting to a percent first and then
smoothing. If, however, one were to compute the overall July mean by
factoring in how many events went into each month, that would be weighted and
would be analogous to our smoothing in number of events and then converting
to a percentage.

Figure 9Long-term evolution of the diurnal sampling bias for three different
data sets. The wider latitude bins are representative of data from
Fig. 8. To remove the influence of the rapid monthly
variability, the data are smoothed over 12 months and converted to a percent
of total events. The left column first converts differences in total number
of SR and SS events to percentages and then smooths, while the right column first
smooths differences in number of events and then converts to a percentage.

Figure 9 illustrates this approach for the three latitude
bands for each of the three main data sets (i.e., SAGE II, HALOE, and
ACE-FTS) where the “unweighted” approach is shown in the left column and the
“weighted” approach is shown in the right column. As can be seen, the
unweighted approach is more susceptible to creating a diurnal sampling bias
that aliases into longer-duration variability than the weighted approach.
However, even the weighted approach reveals that diurnal sampling biases
cannot be avoided. The previously noted SAGE II instrument problem periods
create large diurnal sampling biases with the net effect of creating large
discrepancies in derived potential recovery trends (i.e., post-1997–1998)
between the MZM and STS approaches. However, the diurnal sampling bias for
HALOE appears to have some QBO-like periodicity that is hemispherically
anticorrelated and that for ACE-FTS appears to have an overall trend in the
tropics. For each case, it becomes apparent that even an attempt to account
for the diurnal sampling of these instruments in an MZM analysis (i.e., the
weighted case) will still introduce biases unless the diurnal variability
is specifically modeled or corrected for beforehand.

The nonuniform temporal, spatial, and diurnal sampling patterns present in
occultation instruments detrimentally impact trend results derived from the
MZM method. To illustrate this, we also employ an MZM regression to compare
with the STS regression. The MZM method employed is a one-dimensional (i.e.,
time only) regression that utilizes monthly means with a minimum of 5 events
in 10∘ wide latitude bins without differentiating between sunrise and
sunset events, but otherwise uses the same proxies and statistical analysis
as the STS method. Since the MZM method cannot compensate for the various
sampling biases but is the de facto methodology for data product usage (e.g.,
trend analyses or incorporation into models), we also used the results of the
STS method to create corrected versions of the different data sets for
incorporation into the MZM method. The first is a diurnally corrected data
set that simply applies the derived diurnal variability to bring all
individual sunrise events into the sunset regime. The second applies the
diurnal correction and also uses the spatially varying seasonal ozone
gradient to compute a correction based on the difference between the location
and time an event occurred versus the center of that month and the latitude
bin it would fall within for a particular MZM averaging scheme. It is
important to note that this “seasonal correction” retains variability between
events within a month and bin (i.e., it does not make all values the same)
and is specific to the latitude bin (i.e., width of the bin and center of the
bin). The MZM regression is then applied to each of these three data versions
(i.e., uncorrected, or “Raw”; diurnally corrected, or “DCorr”; and diurnally
and seasonally corrected, or “DSCorr”).

Figure 10Long-term trends derived from both the MZM and the STS regressions
during the typical decline period. Results are also shown when using the STS
regression results to create a diurnally corrected (DCorr) and a diurnally
and seasonally corrected (DSCorr) data set for use with the MZM regression. The
diurnal correction has the greatest influence on the upper stratosphere while
the seasonal correction has the greatest influence at higher latitudes.
Stippling denotes areas where the trend results are not significant at the
2σ level. Contour lines are plotted at 2 % intervals.

We compute trends and uncertainties using the resulting two orthogonal
EESC-proxy functions and the method described in Appendix B.
This uses a simple linear fit to the EESC component of the regression results
evaluated over a desired time period to derive the trend and makes a
correlation between the EESC-fit uncertainties and the functional form of the
linear fit to derive the associated trend uncertainty. The derived trends
from the MZM and STS methods for a typical decline period (1985–1995) are
shown in Fig. 10. As expected, the difference between the MZM
and STS methods during this time is small. The diurnal correction (i.e.,
comparing MZM DCorr with MZM Raw) has some limited impact in the upper
stratosphere at midlatitudes while the seasonal correction (i.e., comparing
MZM DSCorr with MZM DCorr) has larger influence at higher latitudes (at
all altitudes) as well as some minor influence in the tropical middle
stratosphere, though trends in this area are not significant. Overall,
however, the resulting trends are typical of other studies, though we would
like to note that the positive trends in the tropical lower stratosphere
below ∼ 23 km are, similar to the solar cycle
response, detrimentally affected by the anomalous aerosol response.

Figure 11Long-term trends derived from both the MZM and the STS regressions
during the potential recovery period. Results are also shown when using the
STS regression results to create a diurnally corrected (DCorr) and a
diurnally and seasonally corrected (DSCorr) data set for use with the MZM
regression. The diurnal correction has the greatest influence on the upper
stratosphere while the seasonal correction has the greatest influence at
higher latitudes. Stippling denotes areas where the trend results are not
significant at the 2σ level. Contour lines are plotted at 2 %
intervals.

The derived trends from the MZM and STS methods for a potential recovery
period (2000–2012) are shown in Fig. 11. There are significant
differences between the raw MZM results and the STS results most noticeably
from the diurnal sampling biases. Trends in the upper stratosphere at
mid-southern latitudes decrease by ∼ 1 % decade−1 while trends
in the upper stratosphere at mid-northern latitudes increase
by ∼ 1 % decade−1, which is consistent with the expectations
from diurnal sampling biases in the SAGE II data set. The seasonal
correction, as in the decline period, influences the trends at higher
latitudes as well as some minor influence in the tropical middle
stratosphere. It is worth noting that, generally, the fully corrected MZM data
results agree much better, though expectedly not identically given the
different data resolutions and techniques, with the STS results when compared
to the raw MZM results. Overall the results show statistically significant
trends of about 2–3 % decade−1 in isolated parts of the upper stratosphere at
midlatitudes as well as in the tropical middle stratosphere. However, as
discussed in the next section, there are other factors that affect these
results that may indicate these trends are not only statistically
insignificant but potentially biased as well.

One of the biggest issues in every regression technique is the combination of
multicollinearity and orthogonality. Multicollinearity refers to the fact
that the proxies used in the regression are not orthogonal to every other
proxy used and that individual proxies or linear combinations of proxies are
correlated with other proxies. The larger the collinearity between two or
more proxies, the more difficult it is to separate their influences on the
data. Sometimes proxies are sufficiently independent as to be useable, but
when sampled in a particular way (e.g., to match the sampling of a particular
data set) the resulting subsampled proxies exhibit larger collinearity. A
clear example of this is seen in the diurnal and seasonal sampling patterns
of the three instruments. Over their mission lifetimes, the diurnal and
seasonal sampling patterns in SAGE II and HALOE are sufficiently orthogonal
such that the regression can extract both the diurnal variability and
seasonal cycles in each instrument separately. However, this is not the case
for ACE-FTS as its diurnal and seasonal sampling patterns are highly
correlated. Figure 12 illustrates the diurnal variability for
each instrument when the regression allows each instrument to have its own
seasonal cycle. When compared with Fig. 3, the results for
SAGE II and HALOE are the same, illustrating sufficient orthogonality in their
sampling patterns and the fact that their seasonal cycles are essentially the
same as well. However, the results for ACE-FTS lose coherence and agreement
with other studies. It is for this reason that this work made use of a single
seasonal cycle as it allowed SAGE II and HALOE to constrain the seasonal
cycle and thus make the extraction of the diurnal variability in the ACE-FTS
data set possible. Furthermore, the fact that using a single seasonal cycle
allows the independent extraction of diurnal variability that agrees well
with other studies suggests that all three instrument do, in fact, observe
the same seasonal cycle.

Figure 12Same as Fig. 3 except the regression is allowed to
fit different seasonal cycles for each instrument. The lack of orthogonality
between the diurnal and seasonal sampling patterns in ACE-FTS makes it
impossible to differentiate between the two. SAGE II and HALOE remain
unaffected, illustrating sufficient orthogonality and the fact that their
seasonal cycles are essentially the same.

The most recent Scientific Assessment of Ozone Depletion (WMO, 2014)
noted that a primary problem when attempting to derive long-term trends in
ozone when incorporating multiple data sets is that of instrument offsets and
drifts. Given any overlap between two instruments, the offset between
instruments is easily characterizable though many trend analyses are
performed on anomalies and so these offsets are inherently removed. Drifts
between instruments, however, are much more difficult to characterize.
Hubert et al. (2016) performed an extensive analysis of ground and satellite
data sets in an attempt to assess the average drifts present in each
satellite data set relative to the ground network. The results showed that
some instruments were more stable than others (e.g., SAGE II, HALOE, ACE-FTS,
and MLS), though the degree of overlap between the satellite sampling
patterns and the available ground stations did preclude the ability to
determine the full spatial extent of drifts for every instrument (e.g.,
ACE-FTS).

Figure 13The result of the independent drift term used in the regression
showing the relative drift from the SAGE II data for each of the other
instruments. Derived drifts of ∼ 2–3 % decade−1 through most
of the stratosphere for HALOE agree well with earlier studies but the lack of
orthogonality between trend and drift terms during the overlap between the
ACE-FTS and SAGE II missions causes anomalous results.

This work incorporates an offset and a drift term for HALOE and ACE-FTS
relative to SAGE II. The offset terms (not shown) are similar to those found
in other studies comparing these instruments and are not a focus here.
Figure 13 shows the linear drifts relative to SAGE II.
Throughout most of the stratosphere, HALOE shows a negative drift
of ∼ 2–3 % decade−1 relative to SAGE II, which is in good
agreement with other studies
(e.g., Morris et al., 2002; Nazaryan et al., 2005; Hubert et al., 2016).
The drift results from ACE-FTS, however, require a different interpretation.
A quick comparison of the ACE-FTS drifts in Fig. 13 and the STS
recovery trend results in Fig. 11 shows that the patterns in
the drifts somewhat match the patterns in the trends. This suggests that
trends in the ACE-FTS data set are different from those in the SAGE II data
set and highlights another example of the orthogonality problem. Over the
course of the ACE-FTS mission period, the long-term trend terms and the drift
terms are highly correlated, which is not the case for HALOE because HALOE
spans the ozone turnaround time in the late 1990s, creating sufficient
orthogonality between the long-term trend terms and its drift term. This
means that the long-term trends are constrained by SAGE II and HALOE (and an
independent HALOE drift can also be determined), but any difference in what
the ACE-FTS data may suggest the trend is goes entirely into the drift term.
This is further complicated by the fact that ACE-FTS data only have 2 years
of overlap with SAGE II and HALOE. When the regression is run without any
drift term (Fig. 14), the recovery trend results can be
changed by up to ∼ 2 % decade−1, indicating a potential
additional uncertainty originating from possible drift between this
particular combination of data sets (similar to what was shown in
Harris et al., 2015) and that derived recovery trends are sensitive to how
potential drifts are incorporated or accounted for. Overall, the issue of
orthogonality highlights the limitations of regression techniques and
illustrates how it is actually impossible to simultaneously determine both
potential recovery trends and relative instrument drifts using data from only
after the ozone turnaround.

Figure 14Same as the STS results in Fig. 11 except the
regression no longer assumes any kind of drifts between the instruments.
Reduction in potential recovery trends can be as high as
∼ 2 % decade−1 for this particular combination of data sets.

A simultaneous temporal and spatial regression applied to multiple
occultation data sets simultaneously without homogenization has been
presented. The technique allows for a stratospheric ozone trend analysis that
natively compensates for the nonuniform temporal, spatial, and diurnal
sampling patterns of the data sets, and results on data quality, diurnal
variability, response to aerosol, and the solar cycle were shown. The STS
regression shows the natural derivation of the diurnal variability captured
in each instrument and highlights the impact of how the seasonal cycle is
incorporated, revealing that only a single uniform seasonal cycle should be
used for regression analyses. Comparison of the aerosol responses in SAGE II
and HALOE suggests the need to potentially revisit suggested data usage
filtering criteria and the increased temporal extent of data used in the
study helps to separate apparent aerosol and solar cycle responses to reveal
how only a single solar cycle term should be used. Additionally, a detailed
discussion of the nature of the sampling biases reveals how they impact the
retrieval of long-term trends when performing regressions on MZMs causing
differences in potential recovery trends up to
∼ 1 % decade−1, though we also introduce corrected versions
of the data sets for use with MZM methods that apply a first-order sampling
bias correction for use with trend analyses. While these corrected MZM data
sets naturally do not produce identical results as the STS, they are in
better agreement. This study also highlights the limitations inherent in
regression techniques and details how problems with multicollinearity and
lack of orthogonality can impede accurate determination of long-term trends
in ozone.

For future work, we would like to continue to address the topic of drifts and
orthogonality as this study has shown impacts of the drifts on derived trends
of up to ∼ 2 % decade−1. It is currently impossible to
simultaneously determine both potential recovery trends and relative
instrument drifts but it is also impossible to ascertain a global picture of
drifts for every satellite instrument due to lack of necessary coverage
overlaps. That being said, an analysis could be performed where a relatively
stable and long-lived dense sampler (e.g., MLS) is used as the reference
instrument while incorporating all other desired instruments as well. With
sufficient overlap with all other data sets, an STS regression could
determine the globally resolved drifts between the reference instrument and
all other instruments. The derived trends, however, would come only from the
reference instrument but a follow-up analysis where the reference instrument
is compared to the ground network could ascertain its drift and use it as a
transfer standard. In this way, all instruments could be drift-corrected and
then fed into a final STS regression (without a drift term) so that all of
the data are used to constrain the trend.

This work is primarily a continuation and expansion of Damadeo et al. (2014).
That work discusses the application of a simultaneous temporal and spatial
(STS) multiple linear regression (MLR) analysis applied to SAGE II
stratospheric ozone data. This work uses the techniques described in
Damadeo et al. (2014) and expands them to include multiple occultation data
sets. For the sake of brevity and to assist the reader, this appendix will
summarize the methodology and detail how it was expanded to incorporate
multiple data sets.

Occultation instruments provide observations at two distinct latitude bands
each day separated by spacecraft event type (i.e., sunrise or sunset as seen
by the spacecraft). These observations are evenly distributed in longitude
and span about 3∘ in latitude at the highest latitudes to
about 10∘ in latitude in the tropics. The locations of these bands
gradually move from day to day according to the spacecraft's orbit and can
occasionally cross each other. The data for each instrument are averaged
according to these daily zonal bands and are separated by both the local and
spacecraft event types so that each day can produce up to four data points
for a single instrument. When multiple instruments are used, this process is
done separately for each instrument, meaning that, on a given day, it is
possible to have multiple data points at the same latitude from different
instruments feeding into the regression simultaneously.

The regression model applied to all of this averaged data has the following form:

(A1)η(θ,t)=∑i∑jβi,jΘi(θ)Tj(t),

where η is the concentration of O3, Θi(θ) is the
functional form of the latitude dependence (Legendre polynomials in spherical
harmonics), Tj(t) is the functional form of the temporal dependence, and
βi,j are the coefficients of the regression. The Tj(t)
represent all of the typical proxies used in MLR analyses (e.g., QBO, ENSO,
solar) as well as several conditional proxies. Conditional proxies are
simply 0 or some value (typically 1 to make it a binary conditional term)
depending upon whether a condition is met or not for each data point. For
example, the diurnal variability proxy is 0 for every data point that is a
sunset and 1 for every data point that is a sunrise. In this way, the diurnal
coefficient (or rather set of coefficients because there are multiple “i”
values for each “j”) represents the difference between sunrise and sunset
events. Additionally, some proxies are applied separately by adding another
condition. Continuing the diurnal example, the diurnal variability is
actually applied separately for each of the instruments so instead of a
single Tj(t) there are three (one for each instrument). The condition is
a simple logical “AND” between the diurnal condition just described and a
test to see if the data point of interest comes from the instrument to which
the proxy applies. Similarly, there are two mean offset binary conditional
terms (i.e., one for HALOE relative to SAGE II and one for ACE-FTS relative
to SAGE II) and there are two drift conditional terms with forms
Tj(t)=t-t0,j, where t0,j is chosen at the middle of each
instrument's mission period for HALOE and ACE-FTS. This process of creating
conditional proxies can be repeated to apply certain temporal proxies
separately to data points from different data sets (e.g., having a single
seasonal cycle applied to all data sets or having each data set have its
own).

Once the regression is applied, autocorrelation and heteroscedasticity
corrections are applied as detailed in Damadeo et al. (2014). These corrections
use the total and uncorrelated residuals from the regression to improve the
uncertainties in the coefficients that would otherwise be underestimated.
When using multiple data sets, these corrections are applied separately by
first subsetting the residuals to only those from a single instrument,
applying the corrections, and then repeating the process for each instrument.
Applying these corrections separately ignores correlations between the data
sets and their impacts on the uncertainties. However, we believe this to be a
second-order effect as the occurrence of global perturbations is negligible
and the number of coincidences between the occultation instruments is small
when compared to the ensemble.

The goal of this work is determine ozone trends and their uncertainties from
the proxies used in the regression. In the case of a piecewise linear trend
(PWLT) proxy, the trend is simply the coefficient corresponding to that
particular time period (or, in the case of the STS regression, an aggregate
coefficient evaluated at a particular latitude). Unlike a PWLT term, the
EESC-proxy terms (from Damadeo et al., 2014) are comprised of two separate
temporal coefficients and uncertainties with functional shapes that are
nonlinear, making a simple determination of the resulting overall trends and
uncertainties impossible. Instead, we begin by taking the EESC-proxy
component of the fit and its associated uncertainties that have the following
forms:

(B1)y(θ0,t)=CEESC1(θ0)Tj=EESC1(t)+CEESC2(θ0)Tj=EESC2(t)

and

(B2)σy2(θ0,t)=σEESC12(θ0)TEESC12(t)+σEESC22(θ0)TEESC22(t),

where CEESC1,2(θ0) are the aggregate coefficients from
the regression evaluated at a particular latitude θ0 computed as

(B3)CEESC1,2(θ0)=∑iβi,j=EESC1,2Θi(θ0),

with

(B4)σEESC1,22(θ0)=∑iσβi,j=EESC1,22Θi2(θ0),

and TEESC1,2 are the EESC proxies. The equivalent trend is
then computed by performing a simple linear fit to these data over a desired
time period (e.g., 2000–2012) and using the resulting slope as the trend.
The resulting uncertainty in this slope, however, is more complicated because
the uncertainty from the linear fit can vary with the arbitrary number of
points used to create the EESC fit. We have concluded that the best way to
relate a linear fit to the EESC fit was to draw a corollary to the
uncertainties associated with a straight line fit. A linear fit to the
EESC-fit data and their uncertainty have the following forms:

(B5)y′(t)=c0+c1(t-t0)

and

(B6)σy′(t)=σc02+σc12(t-t0)2,

where y′(t) is the best fit to y(θ0,t) and c0 and c1
come from the linear fit but there is no straightforward calculation to compute σc0
and σc1. It is worth noting that, for the linear fit to the EESC
fit, the choice of t0 is arbitrary when we only care about c1. From
these equations, the correlation is made between the linear equation and its
functional uncertainties (i.e., σy′ that are unknown) and the
actual uncertainties from the EESC fit (i.e., σy). From the above
we have

(B7)σc0=σy′(t0),

to which we draw the corollary

(B8)σc0=MINIMUMσy(θ0,t)=σy(θ0,t0)

that yields σc0 and t0. From there, it is simple to look at σc1:

(B9)σc1=σy′2(t)-σc02(t-t0)2,

to which we draw the corollary

(B10)σc1=MEANσy2(θ0,t)-σc02(t-t0)2.

Thus, using a direct correlation between the EESC fit and the functional form
of the linear fit, the uncertainties in the EESC fit can be used to derive a
reasonable estimate for the uncertainty in the fitted slope.

This article is part of the special issue “Quadrennial Ozone
Symposium 2016 – Status and trends of atmospheric ozone (ACP/AMT
inter-journal SI)”. It is a result of the Quadrennial Ozone Symposium 2016,
Edinburgh, United Kingdom, 4–9 September 2016.

The ongoing development, production, assessment, and analysis of SAGE data
sets at NASA Langley Research Center is supported by NASA's Earth Science
Division. The Atmospheric Chemistry Experiment (ACE), also known as SCISAT,
is a Canadian-led mission mainly supported by the Canadian Space Agency and
the Natural Sciences and Engineering Research Council of Canada.

Remsberg, E. and Lingenfelser, G.: Analysis of SAGE II ozone of the middle
and upper stratosphere for its response to a decadal-scale forcing, Atmos.
Chem. Phys., 10, 11779–11790, https://doi.org/10.5194/acp-10-11779-2010,
2010. a

An ozone trend analysis that compensates for sampling biases is applied to sparsely sampled occultation data sets. International assessments have noted deficiencies in past trend analyses and this work addresses those sources of uncertainty. The nonuniform sampling patterns in data sets and drifts between data sets can affect derived recovery trends by up to 2 % decade−1. The limitations inherent to all techniques are also described and a potential path forward towards resolution is presented.

An ozone trend analysis that compensates for sampling biases is applied to sparsely sampled...